Given the initial condition x(0) = b, where

If an initial condition was given for the system above, then the problem becomes an initial value problem and we have to evaluate the constants c1, c2 and c3 in the general solution.

Initial Conditions

Then we compute A x(t) and

compare it to x'(t).

which gives

To verify that x(t) is a solution to

the system x'(t) = A x(t), we first

define the derivative of the

solution

Verification of the solution

The general solution is formed

using the above solutions

We define a third solution using the

above e-vector

Thus, the second e-vector associated

with is

==>

and

Check

Note: Mathcad requires

a redefinition of x(t)

and the general solution

reduces to

the particular

solution

and

Thus,

Compute the reduced row

echelon form of Xb

This gives

Form the augmented matrix

and name it Xb

To solve such a nonhomogeneous system we form the augmented matrix

[x1(0) x2(0) x3(0) | b ]

and compute the row reduced echelon form as follows

Such a system can rewritten as X c = b

where X = [x1(0) x2(0) x3(0) ] and c =

Now, to find c1, c2 and c3 we need to solve the nonhomogeneous linear system

==>

Let

So, we have one free variable

Compute the row reduced

echelon form of

This can be achieved by computing the reduced row echelon form of the coefficient matrix .

To find the e-vector associated with , we need to solve the homogeneous system

Case

Compute the e-values

by finding the roots of the

characteristic equation

Define I as the 3 x 3 identity

matrix

Solution

where

Solve the system of differential equations

This is a example of solving a homogeneous system of differential equations with a non-defective eigenvalue of multiplicity 2.

Problem 9

To get a second e-vector we choose

We use this e-vector to define

the second solution

With these choices we get

the first e-vector associated

with

==>

and

Let

Here we have two free variables v3 and v2 which means that we can generate two linearly independent e-vector associated with .

Compute the row reduced

echelon form of

Again, to compute the e-vector associated with we compute the rref of

Case

we get

Using the definition the first solution

Thus, the e-vector associated with is